Introduction to philosophy

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I'm a mathematics student who attends meetings of my college's philosophical society. It's a small group of people and most of them seem to appreciate my input as an outsider and an honest fool. Most of what they discuss is very accessible, but there is a general canon of philosophers they frequently discuss who I am only familiar with on a Wikipedia/introduction sort of level (most commonly Plato, Aristotle, Descartes, Hume, Kant, Hegel and Wittgenstein). In my college, courses are very specialised; philosophy students take entirely philosophy courses, and mathematics students take entirely mathematics courses.

By undergraduate standards, I'm pretty strong with logic. I have a decent intuition for deductive reasoning having read through many (and written some) mathematical and logical proofs. I know first-order logic up to its completeness theorem, Gödel, Turing+Church, and set theory up to the independence theorems in ZF for AC and CH. I hope to take a logic-related area to graduate level. However, I also want to at least have a rudimentary understanding of philosophy, partly so I can settle in these philosophical circles and partly to develop more of an intuition for my own interests.

I recently spoke to a philosophy lecturer who is more interested in continental philosophy, particularly Gadamer. Obviously our convictions and interests strongly differed but we were still able to relate on a very basic level, and he recommended I read Wittgenstein's Philosophical Investigations. I took a lend of that and the Tractatus Logico-Philosophicus out and have started to read the latter; I'm aware it's much different, but Wittgenstein mentions in his preface to Philosophical Investigations that he would prefer to have his "investigations" read as a contrast to the Tractatus. A brief skim reveals that Wittgenstein does not make many references to other philosophers, and I have a particular interest in logic and language, so it doesn't appear to have been a misguided choice.

What other philosophical texts, especially but not limited to the "canon" I mentioned, should I read? I have been informed that I should also read books on some of these philosophers, and plan to do so with Wittgenstein to see how my interpretation of say the Tractatus differs from that of others - any recommendations along those lines would be welcomed, as well.

Well, it depends on what you want to specialize in.

I am not as focused upon philosophy of logic or language though, so I could at best just look up names for you and tell you that these folks have important works, but I would not have first-hand experience.

Your school likely has a very big interest in abstraction, and some continental influence if that is the list though. Most of those philosophers are important for logic or epistemology, and Hegel is outright rejected by analytic philosophy. Then again, the professor interested in Gadamer is likely a dead give away for something anyway.

If I were to give names, I would give existentialists, ethicists, philosophers of science, philosophers of religion, political philosophers, and stuff like that. That isn't what your school seems that powerfully engaged in I do not think.

I'm not terribly interested in ethics, admittedly. My only non-mathematics course was on ethics, because that was the only philosophy course I took, and while I do not regret the experience I never felt like what I said really had any grounding. Whereas I feel like I have more of an intuitive grasp when discussing logic and applications in mathematics, science and language.

Naïvely speaking I'm somewhat interested in philosophers who take a radically different view to my own present convictions, if only because as Mill put it "he who knows only his own side of the case, knows little of that".

I kind of liked reading about Rawls and huh.....his arch nemesis.... gah, can't remember the name.... Rawls was proning social justice, and his nemesis was telling that the state shouldn't force you to do what you don't want to do (like, if you don't want to risk your life to save someone, it's completly legitimate, if you don't want to wear a helmet when you're riding a bike, completly legitimate, and hey, you can sell your body too if you want! Completly legitimate here too.). Anyways, he was kinda fun.

Hrm, i remembered about Nozick today -.- as i was posting in another thread, oh well, thanks anyways Awesomelyglorious.